DIGITAL MODULATION
TECHNIQUES
Part IV
by
Dr Debashis Adhikari
Reference: 1. Digital Communication by John Proakis, 4th Edition , McGrawHill
2. Digital Communication by John Proakis & Salehi, 5th Edition , McGrawHill
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Quadrature Amplitude Modulation
(QAM)
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2D. Quadrature Amplitude Modulation (QAM)
• Applications: QAM has wide applications in digital cable television, in internet services, in cellular technology, wireless
device technology, wireless microwave links, wired telephone lines etc.
• In QAM, two carrier signals of the same frequency, but in phase quadrature, are independently amplitude modulate
by discrete amplitudes.
• Generalized Equation of QAM
𝑠𝑚 𝑡 = 𝐴𝑚𝑐 𝑔 𝑡 cos 2𝜋𝑓𝑐 𝑡 − 𝐴𝑚𝑠 𝑔 𝑡 sin 2𝜋𝑓𝑐 𝑡 , 𝑚 = 0, 1, 2, . . , 𝑀.
= 𝑅𝑒 𝐴𝑚𝑐 + 𝑗𝐴𝑚𝑠 𝑔 𝑡 𝑒 𝑗2𝜋𝑓𝑐 𝑡 , 𝑚 = 1,2, . . 𝑀
= 𝑅𝑒 𝑉𝑚 𝑒 𝑗𝜃𝑚 𝑔 𝑡 𝑒 𝑗2𝜋𝑓𝑐 𝑡 , 𝑚 = 1,2, . . 𝑀 𝐴𝑚𝑠
𝜃𝑚 = tan−1
𝐴𝑚𝑐
𝑗2𝜋𝑓𝑐 𝑡+𝑗𝜃𝑚
𝑠𝑚 𝑡 = 𝑅𝑒 𝑉𝑚 𝑔 𝑡 𝑒 , 𝑚 = 1,2, . . 𝑀 𝑉𝑚 = 𝐴2𝑚𝑠 + 𝐴2𝑚𝑐
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2D. Quadrature Amplitude Modulation (QAM)
𝑠𝑚 𝑡 = 𝑅𝑒 𝑉𝑚 𝑔 𝑡 𝑒 𝑗2𝜋𝑓𝑐 𝑡+𝑗𝜃𝑚 , 𝑚 = 1,2, . . 𝑀
• From the above expression it is apparent that QAM can be viewed as a combined amplitude and phase modulation.
• Therefore we can select any combination of M1 – ary PAM and M2 – ary PSK to construct an M=M1M2 ary PAM – PSK
signal constellation.
• If the M1 – ary PAM symbols are mapped to 𝑘1 = log 2 𝑀1 binary digits and the M2 – ary PSK to 𝑘2 = log 2 𝑀2 then the
combined QAM constellation results in the simultaneous transmission of 𝑘1 + 𝑘2 = log 2 𝑀 bits per symbol.
• So if bit transmission rate is 𝑅𝑏 bits /sec, then the symbol transmission rate is 𝑅𝑠 = 𝑅𝑠 𝑘1 +𝑘2 symbols /sec.
• Hence the transmission bandwidth also gets reduced proportionately.
• Also transmitting the symbols simultaneously in the quadrature channels using the same carrier frequency is an added
advantage.
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2D. Quadrature Amplitude Modulation (QAM)
The QAM signal can therefore be represented as
𝑠𝑚 𝑡 = 𝑠𝑚1 𝜑1 𝑡 + 𝑠𝑚2 𝜑2 𝑡
Basis functions Coordinates
2
𝜑1 𝑡 = 𝑔 𝑡 cos 2𝜋𝑓𝑐 𝑡
𝐸𝑔
𝐸𝑔 𝐸𝑔
𝐬𝑚 = 𝑠𝑚1 𝑠𝑚2 = 𝐴𝑚𝑐 𝐴𝑚𝑠
2 2
2
𝜑2 𝑡 = 𝑔 𝑡 sin 2𝜋𝑓𝑐 𝑡
𝐸𝑔
Euclidean distance
𝑒 1 2 2
𝑑𝑚𝑛 = 𝐬𝐦 − 𝐬𝐧 = 𝐸 𝐴 − 𝐴𝑛𝑐 + 𝐴𝑚𝑠 − 𝐴𝑛𝑠 = 𝑑 2𝐸𝑔
2 𝑔 𝑚𝑐
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2D. Quadrature Amplitude Modulation (QAM)
Signal waveform for a M- ary Pulse amplitude Modulation (PAM)
𝑠𝑚 𝑡 = 𝑅𝑒 𝐴𝑚 𝑔 𝑡 𝑒 𝑗2𝜋𝑓𝑐 𝑡 = 𝐴𝑚 𝑔 𝑡 cos 2𝜋𝑓𝑐 𝑡, 𝑚 = 1,2, … 𝑀, 0 ≤ 𝑡 ≤ 𝑇𝑠
Signal waveform for a M – ary Phase shift Keying (PSK)
𝑚−1
𝑗2𝜋
𝑠𝑚 𝑡 = 𝑅𝑒 𝑔 𝑡 𝑒 𝑀 . 𝑒 𝑗2𝜋𝑓𝑐 𝑡 , 𝑚 = 1, 2, . . 𝑀, 0 ≤ 𝑡 ≤ 𝑇𝑠
2𝜋 2𝜋
= 𝑔 𝑡 cos 𝑚 − 1 cos 2𝜋𝑓𝑐 𝑡 − 𝑔 𝑡 sin 𝑚 − 1 sin 2𝜋𝑓𝑐 𝑡
𝑀 𝑀
2𝜋
𝑠𝑚 𝑡 = 𝐴𝑚𝑐 𝑔 𝑡 cos 2𝜋𝑓𝑐 𝑡 − 𝐴𝑚𝑠 𝑔 𝑡 sin 2𝜋𝑓𝑐 𝑡 𝐴𝑚𝑐 = cos 𝑚−1
𝑀
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2D. Quadrature Amplitude Modulation (QAM)
Drawing of Constellation Diagram (Signal Space Diagram) – Approach I (Rectangular constellation)
• We consider a special case when 𝑀1 = 𝑀2 = 𝑀 and the in-phase and quadrature phase amplitudes of QAM
waveforms (i.e., 𝐴𝑚𝑐 & 𝐴𝑚𝑠 ) are chosen as the 𝑀 -ary PAM symbol set.
• The amplitudes will in this case discrete values 2𝑚 − 1 − 𝑀 𝑑/2 , 𝑚 = 1, 2, 3, . . 𝑀
𝑠𝑚𝐼 (𝑡) = 𝐴𝑚𝑐 𝑔 𝑡 cos 2𝜋𝑓𝑐 𝑡, 𝑚 = 1,2, … 𝑀, 0 ≤ 𝑡 ≤ 𝑇𝑠 𝑠𝑚𝑄 𝑡 = 𝐴𝑚𝑠 𝑔 𝑡 sin 2𝜋𝑓𝑐 𝑡, 𝑚 = 1,2, … 𝑀, 0 ≤ 𝑡 ≤ 𝑇𝑠
𝜑2 𝑡 = sin 2𝜋𝑓𝑐 𝑡
+3d
00 01 11 10 +d
𝜑1 𝑡 = cos 2𝜋𝑓𝑐 𝑡
-d
-3d -d +d +3d
-3d
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2D. Quadrature Amplitude Modulation (QAM)
Drawing of Constellation Diagram (Signal Space Diagram) – Approach I (Rectangular constellation) 𝜑2 𝑡
𝑠𝑚𝐼 (𝑡) = 𝐴𝑚𝑐 𝑔 𝑡 cos 2𝜋𝑓𝑐 𝑡 + 𝐴𝑚𝑠 𝑔 𝑡 sin 2𝜋𝑓𝑐 𝑡, 𝑚 = 1,2, … 𝑀, 0 ≤ 𝑡 ≤ 𝑇𝑠
00
01
M = 4, m = 2
00 01 11 10
𝜑1 𝑡
11
M = 4, m = 2
10
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2D. Quadrature Amplitude Modulation (QAM)
Drawing of Constellation Diagram (Signal Space Diagram) – Approach I (Rectangular constellation)
𝑠𝑚𝐼 (𝑡) = 𝐴𝑚𝑐 𝑔 𝑡 cos 2𝜋𝑓𝑐 𝑡 + 𝐴𝑚𝑠 𝑔 𝑡 sin 2𝜋𝑓𝑐 𝑡, 𝑚 = 1,2, … 𝑀, 0 ≤ 𝑡 ≤ 𝑇𝑠
𝜑2 𝑡
𝑠𝑚 𝑡 = 𝑠𝑚1 𝜑1 𝑡 + 𝑠𝑚2 𝜑2 𝑡
(+d, +3d)
00
+3d
1 10 0 1 1 0 1 0 1 1 0 0 1 1 1 1
+d
01
-3d -d +d +3d
𝑠𝑚1 → 𝑀1 = 4; 𝑘 = log 2 𝑀1 = 2
00 01 11 10 𝜑1 𝑡
𝑠𝑚2 → 𝑀2 = 4; 𝑘 = log 2 𝑀2 = 2 -d
11
𝑠𝑚 → 𝑀1 𝑀2 = 16; 𝑘 = log 2 𝑀2 𝑀2 = log 2 𝑀1 + log 2 𝑀2
10
(-d, -3d) -3d
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2D. Quadrature Amplitude Modulation (QAM)
𝑠𝑚 𝑡 = 𝐴𝑚𝑐 𝑔 𝑡 cos 2𝜋𝑓𝑐 𝑡 − 𝐴𝑚𝑠 𝑔 𝑡 sin 2𝜋𝑓𝑐 𝑡
M=16 𝜑2 𝑡 𝜑2 𝑡 M=8
00
+3d
+d 01
-3d -d +d +3d -d
00 01 11 10 𝜑1 𝑡 00 01 11 10 𝜑1 𝑡
-d
11
10
-3d
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2D. Quadrature Amplitude Modulation (QAM)
𝜑2 𝑡
Gray coding 000
111000
001
110001 111001 101001
011
111011
010
000 001 011 010 110 111 101 100 𝜑1 𝑡
110
111
101 101
100
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2D. Quadrature Amplitude Modulation (QAM)
Drawing of Constellation Diagram (Signal Space Diagram) – Approach II (Circular constellation)
M=16
𝜑2 𝑡
2𝜋 2𝜋
𝑠𝑚 𝑡 = 𝑔 𝑡 cos 𝑚 − 1 cos 2𝜋𝑓𝑐 𝑡 − 𝑔 𝑡 sin 𝑚 − 1 sin 2𝜋𝑓𝑐 𝑡
𝑀 𝑀
2𝜋 2𝜋
= 𝑔 𝑡 𝐴𝑚𝑐 cos 𝑚 − 1 cos 2𝜋𝑓𝑐 𝑡 − 𝑔 𝑡 𝐴𝑚𝑠 sin 𝑚 − 1 sin 2𝜋𝑓𝑐 𝑡
16 16
𝜑1 𝑡
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2D. Quadrature Amplitude Modulation (QAM)
Choice of Constellation
• We should keep 𝑑 𝑒 as far apart as possible.
• We should select a constellation which has minimum energy.
• We should choose a constellation which exhibits minimum peak to average power
ratio (PAPR).
• We should choose the constellation for such an application which is least affected by
non – linearity problems of any amplifier.
• We should any constellation which provides easy processing at the receiver.
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2D. Quadrature Amplitude Modulation (QAM)
𝑒
M=16 𝜑2 𝑡 𝐸𝑠1 = 𝑑11 = 𝑑 2 + 𝑑 2 =2𝑑 2
𝑒
2d s4(d,3d) s3(3d,3d) 𝐸𝑠2 = 𝑑21 = 9𝑑 2 + 𝑑 2 = 10𝑑 2
+3d 𝑒
𝐸𝑠3 = 𝑑22 = 9𝑑 2 + 9𝑑 2 = 18𝑑 2
s1(d,d) s2(3d,d) 𝑒
+d 𝐸𝑠4 = 𝑑12 = 9𝑑 2 + 𝑑 2 = 10𝑑 2
-3d -d +d +3d
Total energy in first quadrant = 40𝑑 2
𝜑1 𝑡
-d Total energy o𝐟 𝟏𝟔 𝑸𝑨𝑴 ; 𝐸𝑠 = 160𝑑 2
Average energy o𝐟 𝟏𝟔 𝑸𝑨𝑴 ; 𝐸𝑎𝑣 = 10𝑑 2
-3d 𝐸𝑠3 18𝑑2
Peak – to – average power ratio (PAPR) = = = 1.8
𝐸𝑎𝑣 10𝑑2
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2D. Quadrature Amplitude Modulation (QAM)
QAM Transmitter
𝐴𝑚𝑐 𝑡
PAM
I
+
Input Serial to cos 2𝜋𝑓𝑐 𝑡
cos 2𝜋𝑓𝑐 𝑡 QAM o/p
binary Parallel
Converter 𝑠𝑚 𝑡
sequence -
Q 𝐴𝑚𝑠 𝑡
PAM
sin 2𝜋𝑓𝑐 𝑡
sin 2𝜋𝑓𝑐 𝑡
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2D. Quadrature Amplitude Modulation (QAM)
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2D. Quadrature Amplitude Modulation (QAM)
QAM Receiver
𝑇
𝐴𝑚𝑐 𝑡 𝑏1
A/D
𝑑𝑡
Converter
0 𝑏2
cos 2𝜋𝑓𝑐 𝑡 Parallel to
𝑟 𝑡 Serial 𝑏 𝑡
= 𝑠𝑚 𝑡 + 𝑛 𝑡 Converter
𝑇 𝑏3
𝐴𝑚𝑠 𝑡
A/D
𝑑𝑡
Converter 𝑏4
0
sin 2𝜋𝑓𝑐 𝑡
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To be continued
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